Superrigidity for homomorphisms into isometry groups of CAT(−1) spaces

CAT(–1) spaces are generalizations of manifolds with negative curvature. In this paper, we prove three types of rigidity results related to CAT(–1) spaces, namely the rigidity of the isometric actions on CAT(–1) spaces under the commensurability subgroups, the higher rank lattices and certain ergodic cocycles. The main idea for our approach relies on a study of the boundary theory we established for the general CAT(–1) spaces.